Overview:
Polynomial expressions can be added and subtracted just like any other type of expression. However, it is important to make sure that only like terms are combined, and unlike terms are kept separate from each other. The resulting expression contains all the like and unlike terms of all the polynomials being considered.
What Are Like Terms?
Like terms contain the same variable at the same degree. For example, 5y and 6y are like terms because they contain the same variable y. Similarly, the constants 2 and 3 are like terms because they do not contain any variables. Also, 3x2 and 10x2 are also like terms.
What Are Unlike Terms?
Unlike terms do not contain the same variables at the same degree. For example, 5a and 2b cannot be added because they do not contain the same terms. Similarly, x4 and x2 cannot be combined, even though they contain the same variable, because the variable is not to the same degree. If a monomial, 5a, is added to another monomial, 2b, the resulting expression is a polynomial, 5a + 2b. Also, if a monomial, 13x2, is added to another monomial, 2x9, the resulting expression is a polynomial, 2x9 + 13x2.
What About Adding Polynomials?
Suppose one polynomial is 5y2+ 10y + 25, and another one is 8y2 + 6y – 10. There are two steps to adding them. First, it helps to group like terms together, such that (5y2 + 8y2) + (10y + 6y) + (25 – 10). Then, add the like terms, as in (5 + 8)y2 + (10 + 6)y + (25 – 10), using the Distributive Property. The end result is 13y2 + 16y + 15.
What About Subtracting Polynomials?
The only difference when subtracting polynomials is to remember to use the inverse of each term that is being subtracted. Suppose one polynomial is 5z2 + 9z – 30 and the one to subtract is z2 -12z + 18. The like terms will be (5z2 -z 2) + (9z + 12z) + (-30 – 18). The end result will be 4z2 – 3z – 48.
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